§1.2 The notion of limit

Introduction

The Notion of Limit

Instantaneous Velocity

Summary

For the function $g$ pictured in Figure 1.2 from Preview Activity 1.2, what can we say about the values of

?

For the function $g$ pictured in Preview Activity 1.2, what can we say about the values of

?

For the function $g$ pictured in Preview Activity 1.2, what can we say about the values of

?

For each of the following functions, we'd like to know whether or not the function has a limit at the stated $a$-values. Use both numerical and algebraic approaches to investigate and, if possible, estimate or determine the value of the limit. Compare the results with a careful graph of the function on an interval containing the points of interest.

-$f(x) = \frac{4-x^2}{x+2}$; $a = -1$, $a = -2$-$g(x) = \sin\left(\frac{\pi}{x}\right)$; $a = 3$, $a = 0$

$\displaystyle \lim_{x \to 1} \frac{x^2 - 1}{x-1}$

$\displaystyle \lim_{x \to 0} \frac{(2+x)^3 - 8}{x}$

Determine the average velocity of the object on the interval $[3,3.2]$. Include units on your answer.

Determine the instantaneous velocity of the object when $t = 3$. Include units on your answer.

Evaluate $\displaystyle \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$.

Let $g(x) = -\frac{|x+3|}{x+3}$. What can be said about $\displaystyle \lim_{x \to -3} g(x)$?

An object moves along a line with position $s(t) = t^2$ meters at time $t$ minutes. Which expression gives the instantaneous velocity at $t = 3$?

Evaluate $\displaystyle \lim_{x \to 0} \frac{(2+x)^3 - 8}{x}$.

For $s(t) = t^2$ meters with $t$ in minutes, the average velocity on $[3, 3+h]$ simplifies to $6 + h$. Use this to compute the average velocity on $[3, 3.2]$, in meters per minute.

For $s(t) = t^2$, the average velocity on $[3, 3+h]$ simplifies exactly to $6 + h$ for every $h$ with $h \ne 0$.